The area of a square and a rectangular field is equal and is `900 m^2`. If the perimeter of the rectangular field is 2 m more than that of the square field, calculate the dimensions of the rectangular field.

Options

A. 24 m, 10 m
B. 30 m, 25 m
C. 36 m, 25 m
D. 36 m, 24 m

Answer

To solve the problem, we need to find the dimensions of the rectangular field given that the area of both the square and rectangular fields is equal to 900 m², and the perimeter of the rectangular field is 2 m more than that of the square field.

### Step-by-Step Solution:

1. **Find the side length of the square:**
The area of the square is given by the formula:
\[ \text{Area} = \text{side}^2 \]
Given that the area is 900 m², we can set up the equation:
\[ \text{side}^2 = 900 \]
To find the side length, take the square root of both sides:
\[ \text{side} = \sqrt{900} = 30 \text{ m} \]

2. **Calculate the perimeter of the square:**
The perimeter of a square is given by the formula:
\[ \text{Perimeter} = 4 \times \text{side} \]
Substituting the side length we found:
\[ \text{Perimeter} = 4 \times 30 = 120 \text{ m} \]

3. **Determine the perimeter of the rectangular field:**
According to the problem, the perimeter of the rectangular field is 2 m more than that of the square:
\[ \text{Perimeter of rectangular field} = 120 + 2 = 122 \text{ m} \]

4. **Set up the perimeter equation for the rectangular field:**
Let the length of the rectangular field be \( l \) and the width be \( w \). The perimeter of a rectangle is given by:
\[ \text{Perimeter} = 2(l + w) \]
Setting this equal to the perimeter we calculated:
\[ 2(l + w) = 122 \]
Dividing both sides by 2:
\[ l + w = 61 \]

5. **Set up the area equation for the rectangular field:**
The area of the rectangular field is given by:
\[ \text{Area} = l \times w \]
Since the area is also 900 m²:
\[ l \times w = 900 \]

6. **Solve the system of equations:**
Now we have two equations:
1. \( l + w = 61 \)
2. \( l \times w = 900 \)

From the first equation, we can express \( w \) in terms of \( l \):
\[ w = 61 - l \]
Substitute \( w \) into the second equation:
\[ l(61 - l) = 900 \]
Expanding this gives:
\[ 61l - l^2 = 900 \]
Rearranging the equation:
\[ l^2 - 61l + 900 = 0 \]

7. **Use the quadratic formula to solve for \( l \):**
The quadratic formula is given by:
\[ l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -61 \), and \( c = 900 \):
\[ l = \frac{61 \pm \sqrt{(-61)^2 - 4 \times 1 \times 900}}{2 \times 1} \]
Calculating the discriminant:
\[ (-61)^2 = 3721, \quad 4 \times 1 \times 900 = 3600 \]
\[ \sqrt{3721 - 3600} = \sqrt{121} = 11 \]
Now substituting back into the formula:
\[ l = \frac{61 \pm 11}{2} \]
This gives two possible values for \( l \):
\[ l = \frac{72}{2} = 36 \quad \text{or} \quad l = \frac{50}{2} = 25 \]

8. **Find the corresponding values of \( w \):**
If \( l = 36 \):
\[ w = 61 - 36 = 25 \]
If \( l = 25 \):
\[ w = 61 - 25 = 36 \]

Thus, the dimensions of the rectangular field are \( 36 \text{ m} \) and \( 25 \text{ m} \).

### Final Answer:
The dimensions of the rectangular field are \( 36 \text{ m} \) (length) and \( 25 \text{ m} \) (width).